Optimal Consumption and Portfolio Control for Jump–Diffusion Stock Process with Log–Normal Jumps

A computational solution is found for a optimal consumption and portfolio policy problem in which the underlying stock satisfies a geometric jump–diffusion in which both the diffusion and jump amplitude are log–normally distributed. The optimal objective is to maximize the expected, discounted utility of terminal wealth and the cumulative discounted utility of instantaneous consumption. The jump– diffusion allows for a more realistic distribution, skewed toward negative jumps and having leptokurtic behavior in which the tails are thicker so that the distribution is more slender around the peak than normal. Computational issues pertinent to jump–diffusion calculations are discussed.